This paper is concerned with a theoretical and numerical analysis for the stability of a vibrating beam of finite length which is fixed at one end and free at the other end and with a dynamical boundary control. On the theoretical results, we prove the existence and uniquenes of global solutions, and the stability of the total energy. Furthemore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time. Error estimates fot the semi-discrete and fully discrete schemes are provided and numerical experiments performed. From the numerical results, the rate of convergence are shoown to be consistent with the order of convegence expected from the theoretical ones.

本文涉及有限长度的振动梁的稳定性的理论和数值分析，该振动梁的一端固定而另一端自由，并具有动态边界控制。在理论结果上，我们证明了整体解的存在性和唯一性，以及总能量的稳定性。此外，我们引入了一种基于有限元离散化的数值方法，该方法适用于空间变量和有限差分方案。提供了半离散和完全离散方案的误差估计，并进行了数值实验。从数值结果来看，收敛速度应与理论上预期的收敛顺序一致。